Related papers: On the Mathieu Conjecture for $Sp(N)$ and $G_2$

We present an explicit integration formula for the Haar integral on a compact connected Lie group. This formula relies on a known decomposition of a compact connected simple Lie group into symplectic leaves, when one views the group as a…

We begin an investigation of supersymmetric theories based on exceptional groups. The flat directions are most easily parameterized using their correspondence with gauge invariant polynomials. Symmetries and holomorphy tightly constrain the…

An outline of a proof of the decomposition of the linear metric perturbation into gauge-invariant and gauge-variant parts on an arbitrary background spacetime is discussed through an exlicit construction of gauge-invariant and gauge-variant…

We give explicit, practical conditions that determine whether or not a closed, connected subgroup H of G = SU(2,n) has the property that there exists a compact subset C of G with CHC = G. To do this, we fix a Cartan decomposition G = K A K…

We prove Breuil's lattice conjecture for higher Hodge-Tate weights in the case of $\mathrm{GL}_2(K)$ where $K$ is an unramified extension of $\mathbb{Q}_p$. More precisely, under some genericity conditions, we show that the lattice inside a…

Given an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the K-groups of the (reduced) groupoid C*-algebra, provided the groupoid has torsion-free…

We first give a characterization for Mathieu subspaces of univariate polynomial algebras over fields in terms of their radicals. We then deduce that for some classes of classical univariate orthogonal polynomials the Image Conjecture is…

The Moore-Tachikawa conjecture is that each connected complex semisimple group $G$ determines a two-dimensional TQFT in a category of Hamiltonian symplectic varieties. While it would be worthwhile to prove this conjecture outright, our…

In our earlier work, we studied the $\hat{Z}$-invariant(or homological blocks) for $SO(3)$ gauge group and we found it to be same as $\hat{Z}^{SU(2)}$. This motivated us to study the $\hat{Z}$-invariant for quotient groups…

Let $G \subseteq \tilde{G}$ be two quasisplit connected reductive groups over a local field of characteristic zero and $G_{der} = \tilde{G}_{der}$. Although the existence of L-packets is still conjectural in general, it is believed that the…

The van der Waerden's Conjecture states that the set $\mathscr{P}_{n,N}^0(\mathbb{Q})$ of monic integer polynomials $f(X)$ of degree $n$, with height $\le N$ such that the Galois group $G_{K_f/\mathbb{Q}}$ of the splitting field…

The goal of this paper is to better understand a family of linear degenerations of the classical Lagrangian Grassmannians $\Lambda(2n)$. It is the special case for $k=n$ of the varieties $X(k,2n)^{sp}$, introduced in previous joint work…

Brlek and Reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2D(u) = \sum_{n=0}^{\infty}T_u(n) in which D(u) denotes the defect of u and T_u(n) denotes C_u(n+1)-C_u(n) +2 - P_U(n+1) -…

We prove Malle's conjecture for $G \times A$, with $G=S_3, S_4, S_5$ and $A$ an abelian group. This builds upon work of the fourth author, who proved this result with restrictions on the primes dividing $A$.

Martin's Conjecture is a proposed classification of the definable functions on the Turing degrees. It is usually divided into two parts, the first of which classifies functions which are not above the identity and the second of which…

We prove a conjecture of Kumbhakar, Roy, and Srinivasan (2024) on the classification of order one differential equations, and a conjecture of Kumbhakar and Srinivasan (2025) on higher order equations. Both conjectures are shown to be…

The longstanding Alperin weight conjecture and its blockwise version have been reduced to simple groups recently by Navarro, Tiep, Spaeth and Koshitani. Thus, to prove this conjecture, it suffices to verify the corresponding inductive…

The so-called inductive McKay condition on finite simple groups, due to Isaacs-Malle-Navarro (2007), has been recently reformulated by Sp\"ath. We show that this reformulation applies to the reduction theorem for Alperin's weight…

In this paper, from the group-theoretic point of view it is investigated such class of the generalized Kompaneets equations (GKEs): $$u_t=\frac1{x^2}\cdot\left[x^4(u_x+f(u))\right]_x, \ (t,x) \in \mathbb{R}_{+} \times \mathbb{R}_{+},$$…

We study the action of a differential operator on Schubert polynomials. Using this action, we first give a short new proof of an identity of I. Macdonald (1991). We then prove a determinant conjecture of R. Stanley (2017). This conjecture…